I’ve edited the pot file and saved the po and mo files in /wp-content/themes/classico/languages.
But some words where still be shown in english, for example “checkout” in the flyout of the checkout button on the right top corner of the page.
What is my mistake, is there a second language file or place to insert hte mo and po files?

Please read this article https://xstore.helpscoutdocs.com/article/30-base-theme-translation (it is for xstore theme, but for classico the translation works in same way).
Place translations files into the /wp-content/languages/themes/ folder with correct names (your language name and classico theme suffix, e.g classico-de_DE.po, classico-de_DE.mo).
I would recommend you to use Loco Translate plugin, that allows doing synchronizations in the simplest way. Button Sync (http://prntscr.com/glemdq) allows you to synchronize your translation file with updated pot file.
If after uploaded translation in correct folder with correct names and the synchronizations your problem won’t be solved, provide us with temporary wp-admin and FTP access to take a closer look.